# Thermodynamic relations

Notation used:

• ${\displaystyle H}$ is the enthalpy
• ${\displaystyle T}$ is the temperature
• ${\displaystyle G}$ is the Gibbs energy function
• ${\displaystyle U}$ is the internal energy,
• ${\displaystyle V}$ is the volume
• ${\displaystyle A}$ is the Helmholtz energy function
• ${\displaystyle p}$ is the pressure
• ${\displaystyle S}$ is the entropy

These thermodynamic relations are also known as coefficient relations.

${\displaystyle T=\left({\frac {\partial U}{\partial S}}\right)_{V}}$

${\displaystyle p=-\left({\frac {\partial U}{\partial V}}\right)_{S}}$

${\displaystyle T=\left({\frac {\partial H}{\partial S}}\right)_{p}}$

${\displaystyle V=\left({\frac {\partial H}{\partial p}}\right)_{S}}$

${\displaystyle S=-\left({\frac {\partial A}{\partial T}}\right)_{V}}$

${\displaystyle p=-\left({\frac {\partial A}{\partial V}}\right)_{T}}$

${\displaystyle S=-\left({\frac {\partial G}{\partial T}}\right)_{p}}$

${\displaystyle V=\left({\frac {\partial G}{\partial p}}\right)_{T}}$